The AGPL is a free software license. This implies that guarantees the well-known freedoms. Among them, there is the freedom to modify software for your own use.

The AGPL is a free software license. This implies that guarantees the well-known freedoms. Among them, there is the freedom to modify software for your own use. As long as you don't exposed this modified version of a hypothetical AGPL-ed Postgres to the end users, it is considered your own use, where you are free to do anything you want, including using it as the data store for a web application. If you would however expose this modified AGPL-ed Postgres as a DBaaS, in this case you will need to offer as AGPL your changes to the users. But this was not your case. So the answer is clearly NO to both questions.